
Ramanujan's series for \(1/\pi \) : first recorded in Ramanujan's second notebook, proved by Bauer in [1]}, and by Ekhad and Zeilberger using WZ-pairs in [2]}.
For a nice survey on Ramanujan's series, see [3]}.
\(\frac{2}{\pi } = \sum _{k=0}^\infty \frac{4k+1}{(-64)^k} \binom{2k}{k}^3 .\) 

Guillera's series for \(1/\pi ^2\) : found and proved by Guillera in 2002 using WZ-pairs [4]}. For more results on Ramanujan-type series for \(1/\pi ^2\) , see Zudilin's surveys [5]}, [6]}.
\(\frac{128}{\pi ^2} = \sum _{k=0}^\infty (-1)^k \binom{2k}{k}^5 \frac{820k^2 + 180k + 13}{2^{20k}} .\) 

Guillera's Zeilberger-type series for \(\pi ^2\) : found and proved by Guillera using WZ-pairs in [7]}.
\(\frac{\pi ^2}{2} = \sum _{k=1}^\infty \frac{(3k-1)16^k}{k^3\binom{2k}{k}^3}.\) 

Markov–Apéry's series for \(\zeta (3)\) : first discovered by Andrei Markov in 1890, used by Apéry for his irrationality proof, and proved by Zeilberger using WZ-pairs in [8]}.
\(\zeta (3) = \frac{5}{2} \sum _{k=1}^\infty \frac{(-1)^{k-1}}{k^3\binom{2k}{k}}.\) 

Amdeberhan's series for \(\zeta (3)\) : proved by Amdeberhan in 1996 using WZ-pairs [9]}.
\(\zeta (3) = \frac{1}{4} \sum _{k=1}^\infty (-1)^{k-1}\frac{56k^2 - 32k+5}{k^3 (2k-1)^2 \binom{2k}{k}\binom{3k}{k}}.\) 

Bailey–Borwein–Bradley identity: experimentally discovered and proved by Bailey et al. in [10]}, a proof using the Markov-WZ method is given in [11]}, [11]} and its \(q\) -analogue
is presented in [13]}.
\( \sum _{k=0}^\infty \zeta (2k+2) z^{2k} = 3 \sum _{k=1}^\infty \frac{1}{\binom{2k}{k}(k^2-z^2)} \prod _{m=1}^{k-1} \frac{m^2-4z^2}{m^2-z^2}, \quad \text{$z\in  {\mathbb {C}}$ with $|z|<1$}. \) 

van Hamme's supercongruence I: first conjectured by van Hamme [14]}, proved by Mortenson [15]} using \(_6F_5\)  transformations and by Zudilin [16]}
using WZ-pairs.
\(\sum _{k=0}^{\frac{p-1}{2}} \frac{4k+1}{(-64)^k} \binom{2k}{k}^3 \equiv p(-1)^{\frac{p-1}{2}} (\text{mod}~p^3), \) 
where \(p\)  is an odd prime and the multiplicative inverse of \((-64)^k\)  should be computed modulo \(p^3\) .

van Hamme's supercongruence II: first conjectured by van Hamme [14]}, proved by Long [18]} using hypergeometric evaluation identities, one of which is obtained by Gessel using WZ-pairs in [19]}.
\(\sum _{k=0}^{\frac{p-1}{2}}{\frac{6k+1}{256^k}}\binom{2k}{k}^3 \equiv p(-1)^{\frac{p-1}{2}} (\text{mod}~p^4),\) 
where \(p>3\)  is a prime and the multiplicative inverse of \((256)^k\)  should be computed modulo \(p^4\) .

Guo's \(q\) -analogue of van Hamme's supercongruence I: discovered and proved recently by Guo using WZ-pairs in [20]}.
\(\sum _{k=0}^{\frac{p-1}{2}}(-1)^k q^{k^2}[4k+1]_q\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}\equiv [p]_qq^{\frac{(p-1)^2}{4}} (-1)^{\frac{p-1}{2}}\pmod {[p]_q^3},\) 
where for \(n\in  {\mathbb {N}}\) , \((a;q)_n := (1-a)(1-aq)\cdots (1-aq^{n-1})\)  with \((a;q)_0 = 1\) , \([n]_q= 1+q+\cdots + q^{n-1}\)  and \(p\)  is an odd prime.

Hou–Krattenthaler–Sun's \(q\) -analogue of Guillera's Zeilberger-type series for \(\pi ^2\) : inspired by a recent conjecture on supercongruence by Guo in [21]}, and proved using WZ-pairs in [22]}.
This work is also connected to other emerging developments on \(q\) -analogues of series for famous constants and formulae [23]}, [24]}, [25]}.
\(2\sum _{k=0}^\infty q^{2k^2+2k}(1+q^{2k^2+2}-2q^{4k+3}) \frac{(q^2;q^2)_k^3}{(q;q^2)^3_{k+1}(-1; q)_{2k+3}} = \sum _{k=0}^\infty \frac{q^{2k}}{(1-q^{2k+1})^2} . \) 

